3.5 \(\int \frac{(c+d x+e x^2+f x^3)^2}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=320 \[ \frac{2 (a+b x)^{5/2} \left (6 a^2 b^2 \left (2 d f+e^2\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{5 b^7}+\frac{4 (a+b x)^{7/2} \left (10 a^2 b e f-10 a^3 f^2-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{7 b^7}+\frac{4 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7}+\frac{2 \sqrt{a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )^2}{b^7}-\frac{2 (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )}{9 b^7}+\frac{4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7} \]

[Out]

(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x])/b^7 + (4*(b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*(a + b*x)^(3/2))/(3*b^7) + (2*(b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e
+ c*f) + 6*a^2*b^2*(e^2 + 2*d*f))*(a + b*x)^(5/2))/(5*b^7) + (4*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d*e + c*f)
- 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2))/(7*b^7) - (2*(10*a*b*e*f - 15*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)
^(9/2))/(9*b^7) + (4*f*(b*e - 3*a*f)*(a + b*x)^(11/2))/(11*b^7) + (2*f^2*(a + b*x)^(13/2))/(13*b^7)

________________________________________________________________________________________

Rubi [A]  time = 0.243653, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {1850} \[ \frac{2 (a+b x)^{5/2} \left (6 a^2 b^2 \left (2 d f+e^2\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )}{5 b^7}+\frac{4 (a+b x)^{7/2} \left (10 a^2 b e f-10 a^3 f^2-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )}{7 b^7}+\frac{4 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7}+\frac{2 \sqrt{a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )^2}{b^7}-\frac{2 (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )}{9 b^7}+\frac{4 f (a+b x)^{11/2} (b e-3 a f)}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]

[Out]

(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x])/b^7 + (4*(b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*(a + b*x)^(3/2))/(3*b^7) + (2*(b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e
+ c*f) + 6*a^2*b^2*(e^2 + 2*d*f))*(a + b*x)^(5/2))/(5*b^7) + (4*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d*e + c*f)
- 2*a*b^2*(e^2 + 2*d*f))*(a + b*x)^(7/2))/(7*b^7) - (2*(10*a*b*e*f - 15*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)
^(9/2))/(9*b^7) + (4*f*(b*e - 3*a*f)*(a + b*x)^(11/2))/(11*b^7) + (2*f^2*(a + b*x)^(13/2))/(13*b^7)

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{\left (c+d x+e x^2+f x^3\right )^2}{\sqrt{a+b x}} \, dx &=\int \left (\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )^2}{b^6 \sqrt{a+b x}}+\frac{2 \left (b^2 d-2 a b e+3 a^2 f\right ) \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt{a+b x}}{b^6}+\frac{\left (b^4 \left (d^2+2 c e\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (d e+c f)+6 a^2 b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{3/2}}{b^6}+\frac{2 \left (10 a^2 b e f-10 a^3 f^2+b^3 (d e+c f)-2 a b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{5/2}}{b^6}+\frac{\left (-10 a b e f+15 a^2 f^2+b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{7/2}}{b^6}+\frac{2 f (b e-3 a f) (a+b x)^{9/2}}{b^6}+\frac{f^2 (a+b x)^{11/2}}{b^6}\right ) \, dx\\ &=\frac{2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )^2 \sqrt{a+b x}}{b^7}+\frac{4 \left (b^2 d-2 a b e+3 a^2 f\right ) \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) (a+b x)^{3/2}}{3 b^7}+\frac{2 \left (b^4 \left (d^2+2 c e\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (d e+c f)+6 a^2 b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{5/2}}{5 b^7}+\frac{4 \left (10 a^2 b e f-10 a^3 f^2+b^3 (d e+c f)-2 a b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{7/2}}{7 b^7}-\frac{2 \left (10 a b e f-15 a^2 f^2-b^2 \left (e^2+2 d f\right )\right ) (a+b x)^{9/2}}{9 b^7}+\frac{4 f (b e-3 a f) (a+b x)^{11/2}}{11 b^7}+\frac{2 f^2 (a+b x)^{13/2}}{13 b^7}\\ \end{align*}

Mathematica [A]  time = 0.457357, size = 303, normalized size = 0.95 \[ \frac{2 \left (\frac{1}{5} (a+b x)^{5/2} \left (6 a^2 b^2 \left (2 d f+e^2\right )-20 a^3 b e f+15 a^4 f^2-6 a b^3 (c f+d e)+b^4 \left (2 c e+d^2\right )\right )+\frac{2}{7} (a+b x)^{7/2} \left (10 a^2 b e f-10 a^3 f^2-2 a b^2 \left (2 d f+e^2\right )+b^3 (c f+d e)\right )+\frac{2}{3} (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+\sqrt{a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )^2-\frac{1}{9} (a+b x)^{9/2} \left (-15 a^2 f^2+10 a b e f+b^2 \left (-\left (2 d f+e^2\right )\right )\right )+\frac{2}{11} f (a+b x)^{11/2} (b e-3 a f)+\frac{1}{13} f^2 (a+b x)^{13/2}\right )}{b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3)^2/Sqrt[a + b*x],x]

[Out]

(2*((b^3*c - a*b^2*d + a^2*b*e - a^3*f)^2*Sqrt[a + b*x] + (2*(b^2*d - 2*a*b*e + 3*a^2*f)*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*(a + b*x)^(3/2))/3 + ((b^4*(d^2 + 2*c*e) - 20*a^3*b*e*f + 15*a^4*f^2 - 6*a*b^3*(d*e + c*f) + 6*
a^2*b^2*(e^2 + 2*d*f))*(a + b*x)^(5/2))/5 + (2*(10*a^2*b*e*f - 10*a^3*f^2 + b^3*(d*e + c*f) - 2*a*b^2*(e^2 + 2
*d*f))*(a + b*x)^(7/2))/7 - ((10*a*b*e*f - 15*a^2*f^2 - b^2*(e^2 + 2*d*f))*(a + b*x)^(9/2))/9 + (2*f*(b*e - 3*
a*f)*(a + b*x)^(11/2))/11 + (f^2*(a + b*x)^(13/2))/13))/b^7

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Maple [A]  time = 0.007, size = 447, normalized size = 1.4 \begin{align*}{\frac{6930\,{f}^{2}{x}^{6}{b}^{6}-7560\,a{b}^{5}{f}^{2}{x}^{5}+16380\,{b}^{6}ef{x}^{5}+8400\,{a}^{2}{b}^{4}{f}^{2}{x}^{4}-18200\,a{b}^{5}ef{x}^{4}+20020\,{b}^{6}df{x}^{4}+10010\,{b}^{6}{e}^{2}{x}^{4}-9600\,{a}^{3}{b}^{3}{f}^{2}{x}^{3}+20800\,{a}^{2}{b}^{4}ef{x}^{3}-22880\,a{b}^{5}df{x}^{3}-11440\,a{b}^{5}{e}^{2}{x}^{3}+25740\,{b}^{6}cf{x}^{3}+25740\,{b}^{6}de{x}^{3}+11520\,{a}^{4}{b}^{2}{f}^{2}{x}^{2}-24960\,{a}^{3}{b}^{3}ef{x}^{2}+27456\,{a}^{2}{b}^{4}df{x}^{2}+13728\,{a}^{2}{b}^{4}{e}^{2}{x}^{2}-30888\,a{b}^{5}cf{x}^{2}-30888\,a{b}^{5}de{x}^{2}+36036\,{b}^{6}ce{x}^{2}+18018\,{b}^{6}{d}^{2}{x}^{2}-15360\,{a}^{5}b{f}^{2}x+33280\,{a}^{4}{b}^{2}efx-36608\,{a}^{3}{b}^{3}dfx-18304\,{a}^{3}{b}^{3}{e}^{2}x+41184\,{a}^{2}{b}^{4}cfx+41184\,{a}^{2}{b}^{4}dex-48048\,a{b}^{5}cex-24024\,a{b}^{5}{d}^{2}x+60060\,{b}^{6}cdx+30720\,{a}^{6}{f}^{2}-66560\,{a}^{5}bef+73216\,{a}^{4}{b}^{2}df+36608\,{a}^{4}{b}^{2}{e}^{2}-82368\,{a}^{3}{b}^{3}cf-82368\,{a}^{3}{b}^{3}de+96096\,{a}^{2}{b}^{4}ce+48048\,{a}^{2}{b}^{4}{d}^{2}-120120\,a{b}^{5}cd+90090\,{c}^{2}{b}^{6}}{45045\,{b}^{7}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x)

[Out]

2/45045*(b*x+a)^(1/2)*(3465*b^6*f^2*x^6-3780*a*b^5*f^2*x^5+8190*b^6*e*f*x^5+4200*a^2*b^4*f^2*x^4-9100*a*b^5*e*
f*x^4+10010*b^6*d*f*x^4+5005*b^6*e^2*x^4-4800*a^3*b^3*f^2*x^3+10400*a^2*b^4*e*f*x^3-11440*a*b^5*d*f*x^3-5720*a
*b^5*e^2*x^3+12870*b^6*c*f*x^3+12870*b^6*d*e*x^3+5760*a^4*b^2*f^2*x^2-12480*a^3*b^3*e*f*x^2+13728*a^2*b^4*d*f*
x^2+6864*a^2*b^4*e^2*x^2-15444*a*b^5*c*f*x^2-15444*a*b^5*d*e*x^2+18018*b^6*c*e*x^2+9009*b^6*d^2*x^2-7680*a^5*b
*f^2*x+16640*a^4*b^2*e*f*x-18304*a^3*b^3*d*f*x-9152*a^3*b^3*e^2*x+20592*a^2*b^4*c*f*x+20592*a^2*b^4*d*e*x-2402
4*a*b^5*c*e*x-12012*a*b^5*d^2*x+30030*b^6*c*d*x+15360*a^6*f^2-33280*a^5*b*e*f+36608*a^4*b^2*d*f+18304*a^4*b^2*
e^2-41184*a^3*b^3*c*f-41184*a^3*b^3*d*e+48048*a^2*b^4*c*e+24024*a^2*b^4*d^2-60060*a*b^5*c*d+45045*b^6*c^2)/b^7

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Maxima [A]  time = 0.991374, size = 675, normalized size = 2.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(45045*sqrt(b*x + a)*c^2 + 858*c*(35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + 7*(3*(b*x + a)^(5/2)
- 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2 + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x +
a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*f/b^3) + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x +
a)*a^2)*d^2/b^2 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3
/2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4 + 286*(35*(b*x + a)^(9/2)*f + 45*(b*e - 4*a*f)*(b*x + a)^(7/2) - 189*
(a*b*e - 2*a^2*f)*(b*x + a)^(5/2) + 105*(3*a^2*b*e - 4*a^3*f)*(b*x + a)^(3/2) - 315*(a^3*b*e - a^4*f)*sqrt(b*x
+ a))*d/b^4 + 130*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/
2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*e*f/b^5 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)
^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3
/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*f^2/b^6)/b

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Fricas [A]  time = 1.26904, size = 1000, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (3465 \, b^{6} f^{2} x^{6} + 45045 \, b^{6} c^{2} - 60060 \, a b^{5} c d + 24024 \, a^{2} b^{4} d^{2} + 18304 \, a^{4} b^{2} e^{2} + 15360 \, a^{6} f^{2} + 630 \,{\left (13 \, b^{6} e f - 6 \, a b^{5} f^{2}\right )} x^{5} + 35 \,{\left (143 \, b^{6} e^{2} + 120 \, a^{2} b^{4} f^{2} + 26 \,{\left (11 \, b^{6} d - 10 \, a b^{5} e\right )} f\right )} x^{4} + 10 \,{\left (1287 \, b^{6} d e - 572 \, a b^{5} e^{2} - 480 \, a^{3} b^{3} f^{2} + 13 \,{\left (99 \, b^{6} c - 88 \, a b^{5} d + 80 \, a^{2} b^{4} e\right )} f\right )} x^{3} + 3 \,{\left (3003 \, b^{6} d^{2} + 2288 \, a^{2} b^{4} e^{2} + 1920 \, a^{4} b^{2} f^{2} + 858 \,{\left (7 \, b^{6} c - 6 \, a b^{5} d\right )} e - 52 \,{\left (99 \, a b^{5} c - 88 \, a^{2} b^{4} d + 80 \, a^{3} b^{3} e\right )} f\right )} x^{2} + 6864 \,{\left (7 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d\right )} e - 416 \,{\left (99 \, a^{3} b^{3} c - 88 \, a^{4} b^{2} d + 80 \, a^{5} b e\right )} f + 2 \,{\left (15015 \, b^{6} c d - 6006 \, a b^{5} d^{2} - 4576 \, a^{3} b^{3} e^{2} - 3840 \, a^{5} b f^{2} - 1716 \,{\left (7 \, a b^{5} c - 6 \, a^{2} b^{4} d\right )} e + 104 \,{\left (99 \, a^{2} b^{4} c - 88 \, a^{3} b^{3} d + 80 \, a^{4} b^{2} e\right )} f\right )} x\right )} \sqrt{b x + a}}{45045 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*b^6*f^2*x^6 + 45045*b^6*c^2 - 60060*a*b^5*c*d + 24024*a^2*b^4*d^2 + 18304*a^4*b^2*e^2 + 15360*a^
6*f^2 + 630*(13*b^6*e*f - 6*a*b^5*f^2)*x^5 + 35*(143*b^6*e^2 + 120*a^2*b^4*f^2 + 26*(11*b^6*d - 10*a*b^5*e)*f)
*x^4 + 10*(1287*b^6*d*e - 572*a*b^5*e^2 - 480*a^3*b^3*f^2 + 13*(99*b^6*c - 88*a*b^5*d + 80*a^2*b^4*e)*f)*x^3 +
3*(3003*b^6*d^2 + 2288*a^2*b^4*e^2 + 1920*a^4*b^2*f^2 + 858*(7*b^6*c - 6*a*b^5*d)*e - 52*(99*a*b^5*c - 88*a^2
*b^4*d + 80*a^3*b^3*e)*f)*x^2 + 6864*(7*a^2*b^4*c - 6*a^3*b^3*d)*e - 416*(99*a^3*b^3*c - 88*a^4*b^2*d + 80*a^5
*b*e)*f + 2*(15015*b^6*c*d - 6006*a*b^5*d^2 - 4576*a^3*b^3*e^2 - 3840*a^5*b*f^2 - 1716*(7*a*b^5*c - 6*a^2*b^4*
d)*e + 104*(99*a^2*b^4*c - 88*a^3*b^3*d + 80*a^4*b^2*e)*f)*x)*sqrt(b*x + a)/b^7

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Sympy [A]  time = 117.588, size = 1365, normalized size = 4.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**3+e*x**2+d*x+c)**2/(b*x+a)**(1/2),x)

[Out]

Piecewise((-(2*a*c**2/sqrt(a + b*x) + 4*a*c*d*(-a/sqrt(a + b*x) - sqrt(a + b*x))/b + 4*a*c*e*(a**2/sqrt(a + b*
x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2 + 2*a*d**2*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*
x)**(3/2)/3)/b**2 + 4*a*c*f*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2
)/5)/b**3 + 4*a*d*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**
3 + 4*a*d*f*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a
+ b*x)**(7/2)/7)/b**4 + 2*a*e**2*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(
a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**4 + 4*a*e*f*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(
a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**5 + 2*a*f**2*(a*
*6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)**(5/2) - 15*a**2*(a + b*x
)**(7/2)/7 + 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**6 + 2*c**2*(-a/sqrt(a + b*x) - sqrt(a + b*x)) +
4*c*d*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b + 4*c*e*(-a**3/sqrt(a + b*x) - 3*a**2*s
qrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**2 + 2*d**2*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b
*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**2 + 4*c*f*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**
2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3 + 4*d*e*(a**4/sqrt(a + b*x) + 4*a**3*sq
rt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3 + 4*d*f*(-a**5/sqrt(
a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/
7 - (a + b*x)**(9/2)/9)/b**4 + 2*e**2*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3
- 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**4 + 4*e*f*(a**6/sqrt(a + b*x) + 6
*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)**(5/2) - 15*a**2*(a + b*x)**(7/2)/7 + 2*a*(a
+ b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**5 + 2*f**2*(-a**7/sqrt(a + b*x) - 7*a**6*sqrt(a + b*x) + 7*a**5*(a
+ b*x)**(3/2) - 7*a**4*(a + b*x)**(5/2) + 5*a**3*(a + b*x)**(7/2) - 7*a**2*(a + b*x)**(9/2)/3 + 7*a*(a + b*x)*
*(11/2)/11 - (a + b*x)**(13/2)/13)/b**6)/b, Ne(b, 0)), ((c**2*x + c*d*x**2 + e*f*x**6/3 + f**2*x**7/7 + x**5*(
2*d*f + e**2)/5 + x**4*(2*c*f + 2*d*e)/4 + x**3*(2*c*e + d**2)/3)/sqrt(a), True))

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Giac [A]  time = 1.11509, size = 697, normalized size = 2.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(b*x + a)*c^2 + 30030*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*c*d/b + 3003*(3*(b*x + a)^(5/2)
- 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*d^2/b^2 + 6006*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*
sqrt(b*x + a)*a^2)*c*e/b^2 + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt
(b*x + a)*a^3)*c*f/b^3 + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x
+ a)*a^3)*d*e/b^3 + 286*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)
^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*d*f/b^4 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^
(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4 + 130*(63*(b*x + a)^(11/2) - 385*(b*x + a
)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*
a^5)*f*e/b^5 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^
(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*f^2/b^6)/b